 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem :: Theorem 4 (L)
  for R1, R2, R being non empty RelStr,
      X being Subset of R,
      X1 being Subset of R1,
      X2 being Subset of R2 st
    R = Meet (R1,R2) & X = X1 & X = X2 &
    the carrier of R1 = the carrier of R2 holds
      LAp X1 \/ LAp X2 c= LAp X
  proof
    let R1, R2, R be non empty RelStr,
        X be Subset of R,
        X1 be Subset of R1,
        X2 be Subset of R2;
    assume
A1: R = Meet (R1,R2) & X = X1 & X = X2 &
    the carrier of R1 = the carrier of R2;
SS: the InternalRel of R =
      (the InternalRel of R1) /\ the InternalRel of R2 by A1,DefMeet;
sa: dom LAp R1 = bool the carrier of R1 by FUNCT_2:def 1;
sw: dom LAp R2 = bool the carrier of R2 by FUNCT_2:def 1;
aa: the carrier of R =
      (the carrier of R1) /\ the carrier of R2 by A1,DefMeet;
    reconsider XX1 = X as Subset of R1 by A1;
    reconsider XX2 = X as Subset of R2 by A1;
    reconsider XX = X as Subset of R;
    LAp X1 \/ LAp X2 c= LAp X
    proof
      let x be object;
      assume x in LAp X1 \/ LAp X2; then
      per cases by XBOOLE_0:def 3;
      suppose
S2:     x in LAp X1;
        LAp R1 cc= LAp R by aa,A1,Prop16L,SS,XBOOLE_1:17; then
        (LAp R1).XX1 c= (LAp R).XX by sa,ALTCAT_2:def 1; then
        (LAp R1).XX1 c= LAp XX by ROUGHS_2:def 10; then
        LAp XX1 c= LAp XX by ROUGHS_2:def 10;
        hence thesis by S2,A1;
      end;
      suppose
S2:     x in LAp X2;
        LAp R2 cc= LAp R by aa,Prop16L,A1,SS,XBOOLE_1:17; then
        (LAp R2).XX2 c= (LAp R).XX by sw,ALTCAT_2:def 1; then
        (LAp R2).XX2 c= LAp XX by ROUGHS_2:def 10; then
        LAp XX2 c= LAp XX by ROUGHS_2:def 10;
        hence thesis by S2,A1;
      end;
    end;
    hence thesis;
  end;
